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I’ve also heard it called “jerk”.

Oops, that's what I meant to say! So much for my correction.

I actually heard, many years ago, the higher derivatives referred to as "jerk" (3rd, as you said), "jolt" (4th) and "jounce" (5th). But that contradicts the Wikipedia article that says "jounce" is 4th derivative.

Oops, yeah you’re right!

It can help because the higher derivatives also tend to promote vibrations in the system, but I doubt it'd be perceptible by people. I have heard of 5th-order smooth curves being used for very sensitive structures, like the movement of big observatory telescopes.